I have to use a binomial expansion to evaluate $1/\sqrt{4.2}$ to $5$ decimal places. The answer from a calculator is $0.48795$ but I get $0.48202$, so I'm doing something wrong. I've also checked my calculations on a spreadsheet so the problem is with my technique, not arithmetic. Here's what I'm doing:
Using
$$ (x + y)^{-n} = x^{-n}\sum_{k=0}^\infty {^{-n}}C_k \left( \frac y x \right)^k $$ I need $$ (4 + 0.2)^{-1/2} = 4 ^{-1/2} \sum_{k=0}^\infty {^{-n}}C_k \left( \frac {0.2} {4} \right)^k $$
$$ = \frac1 2 \sum_{k=0}^\infty {^{-n}}C_k \left( \frac {1} {2} \right)^k \left( \frac {1} {10} \right)^k $$ Using the recurrence relation $$ ^nC_{k+1} = \frac{n-k}{k+1} {^n}C_k $$ I caluculate
$^{-1/2}C_0=1$;
$^{-1/2}C_1=-3/4$;
$^{-1/2}C_2=5/8$;
$^{-1/2}C_3=-35/64$
So the evaluation should be: $$ \frac 1 2\left( 1 + \frac {-3}{4}.\frac1 2.\frac 1 {10} + \frac {5}{8}.\frac1 4.\frac 1 {100} + \frac {-35}{64}.\frac1 8.\frac 1 {1000} \right) $$
but this is incorrect, as described above. It is enough terms because the last one is $-0.0000342$.
What am I doing wrong?